CDF of Order Statistics

Question

Why when finding the CDF of Yn and Zn do you find P(Y_n<=y) as opposed to P(Y_n<=y_n)? Similarly with Z_n.

Answer 1

The cdf of $Y_n$ can be called $F_{Y_n}(w)$, or $F_{Y_n}(t)$, or $F_{Y_n}(y)$, or $F_{Y_n}(y)$, or $G(y)$, it doesn't matter. If for example we like the variable symbol $t$, we can say $$F_{Y_n}(t)=\Pr(Y_n\le t).$$ It might have been nice to call the variable by the name $y_n$, but that is perhaps a needlessly complicated name for a variable.

Answer 2

All CDFs are functions of a real variable, not just limited to values of the RV. For discrete RVs, the CDF will jump at each of the discrete values by the amount of the probability mass associated with that value.

For example, if you have a RV X representing "heads" and "tails", it might assume the values $0$ and $1$ with probability $\frac12$ each. The CDF is

$$F_X(x) = \begin{cases} 0 & x<0\\ \frac12 & 0\leq x < 1 \\ 1 & x \geq 1 \end{cases} $$

so $F_X(-36)=0$, $F_X(\frac2{17})=\frac12$, and $F_X(\frac{12345}{12344})=1$.